In short
Arab-Sasanian and early Islamic coins are the money of Iran between the Arab conquest of 651 CE and the high caliphal period: first Sasanian-looking silver drachms with Arabic marginal phrases, then, after the reform of Abd al-Malik in 696–697 CE, purely epigraphic dirhams and gold dinars with no images at all.
The transitional coinage
The Arab conquest did not change the money. Governors in Basra, Fars and Khurasan carried on striking Sasanian-type drachms — crowned bust, fire altar, Pahlavi legend — because that was what the population accepted. The only immediate change is a short Arabic phrase in the obverse margin, most often bismillah (in the name of God).
These Arab-Sasanian issues are the most legible record anywhere of a state changing hands without a monetary collapse. Names of the new Arab governors appear in Pahlavi script, dates run in the Yazdgerd era, and the whole design remains Zoroastrian in imagery while the authority behind it is Muslim.
The reform of Abd al-Malik
In 696–697 CE (AH 77–78) the caliph Abd al-Malik replaced the whole system with a purely epigraphic coinage. The image of the ruler and the fire altar disappear; both faces carry Qur'anic text and the declaration of faith, arranged as a central legend inside a circular marginal legend.
The reformed silver dirham weighs about 2.9 g on a broad thin flan; the gold dinar sits near 4.25 g. The marginal legend gives the mint and the AH year written out in words — which makes the entire Islamic series precisely datable and mint-attributable, and is why it is one of the most systematically collected fields in world numismatics.
Iranian regional dynasties
As caliphal control loosened, Iranian dynasties struck in their own names while formally acknowledging the caliph: the Tahirids at Nishapur and Ray, the Saffarids in Sistan, the Samanids at Samarkand, Bukhara and Nishapur, the Buyids in Fars and Iraq.
Samanid dirhams are the classic Silk Road coin — struck in enormous quantity and exported north, they are found in Viking hoards from Sweden to Ireland. A Samanid dirham of Nishapur in a Gotland hoard is the single clearest object lesson in early medieval trade reach.
How to read and value them
Start with the marginal legend on the reverse: it reads "in the name of God, this dirham was struck at [mint] in the year [year written in words]". Learning ten mint names and the Arabic numerals-as-words covers most of the market.
Common Umayyad and Abbasid dirhams in fine condition are inexpensive. Value rises with scarce mints, first-year issues, complete unclipped flans and full legible margins. Gold dinars of any early caliph are expensive; Arab-Sasanian pieces naming rare governors are the specialist's target.